Month: July 1999

How do you distribute drinking water fairly across an area recently hit by a natural disaster? Or how can you make sure you have enough bottles of water, granola bars and fruit in your delivery van to refill all the vending machines at a school when you don’t know how full they are?

Eindhoven University of Technology researcher Natasja Sluijk has developed mathematical models to address these challenges in transportation planning. On Thursday 23 November she successfully defended her dissertation at the Department of Industrial Engineering & Innovation Sciences.

Sluijk obtained her master’s degree at Erasmus University in the field of Operations Research, an area of research focused on the application of mathematical methods in order to optimize processes.

“I’ve always been interested in mathematics and I decided I wanted to do something with it,” she says. On top of that, her father and grandfather, both of whom used to be truck drivers, fueled her interest in transportation and logistics. “That’s how the seed was planted.” The Ph.D. candidate is also very intrigued by uncertainty. “Well, in my research that is, not in my life,” she adds with a laugh. Her doctoral research is where these worlds meet.

Reducing emissions

Her dissertation can be divided into two parts. The first part focuses on so-called two-echelon distribution. “First, you transport the goods in big trucks, because that way you can take many items at once, so you need fewer drivers and you reduce the costs,” she explains.

However, due to environmental zones and emissions regulations, trucks cannot enter cities, which is why smaller vehicles take over the goods at the city limits and bring them to their final destination. These include bicycle couriers or electric vans, which are smaller and more compact.

By dividing the distribution chain into two steps, you can keep costs low while still complying with regulations. Not only does the use of greener vehicles in cities reduce emissions, it also reduces noise pollution and parking problems. “These are the reasons why more and more research is being conducted on two-echelon distribution, on how to optimize it and how to plan routes efficiently,” says Sluijk.

Customer demand uncertainty

The primary focus of her doctoral research is dealing with uncertain customer demand. Normally, a route plan is drawn up for a set of customers with known locations and demands. But what if you don’t know in advance exactly how much you need to deliver?

Sluijk did not include home package deliveries in her research, but rather focused on deliveries from companies to other companies, the so-called B2B market. “Think, for example, of deliveries to locations that require product restocking, such as vending machines,” she explains.

“What you can see in advance is how much has been sold, but it’s only when you arrive at the vending machine that you can see the current demand. Basically, between the time of planning and the time of delivery, the demand can change.” As such, the challenge here is to meet all demands without being left with a surplus of goods.

Sluijk has developed exact mathematical models and algorithms that allow for better handling of uncertain customer demand and optimal route planning solutions within a two-echelon distribution. This enables us to improve the structure of the two-echelon distribution system, making it more sustainable and cost-efficient.

“The most optimal solution ultimately depends on the company’s exact goals,” she emphasizes. Do they want as many satisfied customers as possible or do they prioritize low costs? The mathematical models make it possible to calculate different scenarios and, for example, accurately assess how enhancing customer service affects costs.

Fair distribution

In the second part of her dissertation, she focuses on situations where the total demand exceeds the capacity, in other words, the amount you can supply. Besides cost and efficiency, fairness is another important consideration here.

“For example, I arrive at a customer who asks for eight items, but I decide to supply only six so that I have enough left for the other customers in the delivery route. If I don’t do this, I disadvantage the customers later in the route,” she explains.

The key question here is: how do you ensure a fair distribution of goods when the customer demand is uncertain? Sluijk developed mathematical models that ensure everyone is treated equally. “This is something that has to be done proportionally, because if a customer asks for a hundred items, supplying one fewer item is much less of an issue than if they asked for only five items. So that’s how we factor that in,” she explains.

Humanitarian organizations

The models are applicable not only in B2B supply chains, but also in non-commercial sectors, such as humanitarian organizations. “Suppose there has been a natural disaster and you need to deliver water to different locations, but you don’t know exactly how much to deliver to each location,” she says.

“The same thing applies to food banks; they often collect the food at a central location and then distribute it among the regions.” In these situations, it is crucial to fairly distribute the available resources between the different locations.

Here, the exact methods she has developed can be of great help. “However, we still need to bridge the gap between theory and practice; but in principle, the models are widely applicable and provide a good starting point in the search for desirable solutions. Not only do mathematical models help you arrive at solutions, they also allow you to properly substantiate the decisions made. That is the most transparent approach and also prevents arguments,” she concludes.

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Credit of the article given to Eindhoven University of Technology

Mathematics is at the heart of modern science but we shouldn’t forget other ways to reason, says author and researcher Roland Ennos.

“Science is written in the language of mathematics,” proclaimed Galileo in 1623. And over the past few centuries science has become ever more mathematical. Nowadays, mathematics seems to hold total hegemony, particularly in the fields of quantum physics and relativity – the teaching of modern physics seems to involve deriving an endless series of equations.

But though it is an important tool, mathematical analysis is not the only way of approaching scientific enquiry. Scientists also need to develop concepts on which to build the mathematics and carry out experiments to test and demonstrate their ideas. And they also need to translate the equations back into physical concepts and verbal explanations to make them comprehensible. These other aspects have long been undervalued – in both the teaching and practice of physics – and this has damaged and is continuing to damage our understanding of the world around us.

Nowhere is this better exemplified than in the science of rotation and spin, which might at first glance appear to be a shining example of the triumph of mathematics. In his 1687 magnum opus Principia, Isaac Newton laid out the mathematical workings of our solar system: he showed how the laws of motion and gravity explain how the planets orbit around the sun, and how the spin of the earth causes it to bulge, drives the tides and makes its tilted axis slowly wobble. Over the next hundred years, Newton’s analysis was extended and translated into modern mathematical language. All the problems of cosmology appeared to have been solved, the first of many occasions when scientists have mistakenly thought they had uncovered all the secrets of the universe.

Yet Newton’s triumph was only made possible by his more down-to-earth contemporary Robert Hooke. It was Hooke who made the conceptual leap that an object moving in a circle is travelling at a constant speed but is also accelerating at right angles towards the centre of the circle. He also went on to show experimentally how a universal gravity could provide the force that causes the planets to orbit around the sun and the moon around Earth. He hung a large ball, representing Earth, from the ceiling and a small ball, representing the moon, from the large ball, before pulling them away from vertical and setting them moving. The tension in the ropes, representing gravity, provided the inward force that kept them travelling around in a circle.

Unfortunately, Newton, who came to dominate world science, had little time for such conceptual and experimental approaches, insisting that equations were the only way to describe physical reality. His influence impeded further conceptual advances in mechanics and consequently progress in cosmology. For instance, it delayed our understanding of how the solar system was created.

The accepted model – the nebular hypothesis – was put forward in the 18th century by such luminaries as the philosopher Immanuel Kant and the mathematician Pierre-Simon Laplace. The hypothesis proposed that the solar system formed from a spinning ball of dust and gas. Gravity flattened the ball into a disc before the attraction between the particles pulled them together into planets and moons, all orbiting in the same plane and in the same direction.

All seemed well until the 1850s when engineers such as William Rankine finally developed a new mechanical concept – the conservation of angular momentum – 150 years after the conservation of linear momentum had been accepted. This new concept revealed a potential flaw in the nebular hypothesis that had remained hidden in Newton’s equations. To have shrunk to its size and to spin so slowly, the sun must have lost almost all its angular momentum, something that seemed to break this new law of nature.

It was only 40 years ago that a convincing explanation was proposed about how the sun lost its angular momentum. The charged particles shot out by the sun in the solar wind are channelled within magnetic fields before being flung out slowing the spin of the material that remained and allowing gravity to draw it inwards. It was only two years ago that this explanation was finally verified by the Parker Solar Probe, which found that the solar particles were channelled up to 32 million kilometres outwards before being released. And only in October 2023 did the James Webb Space Telescope reveal the same process occurring in the newly forming solar system of the star HH212.

The overreliance on mathematics also delayed our understanding of how the spin of Earth makes it habitable. By the end of the 18th century, Laplace had derived equations describing how Earth’s spin deflects bodies of water moving over its surface. However, even he failed to observe that it would also affect solid objects and gases, so his work was ignored by the early meteorologists.

This only changed in 1851, when the French physicist Jean Foucault produced a free-hanging pendulum that demonstrated Laplace’s forces in action. The forces diverted the bob to the right during each sweep so that its plane of swing gradually rotated, like a Spirograph drawing. Not only did this prove the spin of Earth to a sceptical public, but it showed schoolteacher William Ferrel that Laplace’s forces would also deflect air masses moving around Earth’s surface. This would explain how global air currents are deflected east and west to form the three convection cells that cover each hemisphere and create the world’s climate zones, and how they divert winds into rotating weather systems, creating depressions, hurricanes and anticyclones. Modern meteorology was born.

In 1835, the French engineer Gaspard-Gustave de Coriolis produced more general equations describing the forces on bodies moving within a rotating reference frame. However, since these were in a paper examining the efficiency of water wheels, his work was largely ignored by scientists. Instead, it was a simple experiment that enabled geophysicists to understand how Earth’s spin diverts fluid movements in its interior and produces its magnetic field.

In 1911, the British physicist G. I. Taylor investigated how beakers of water behave when they are set spinning. The water quickly spins with the beaker and its surface rises in a parabola until the extra pressure counters the centrifugal force on the water. What’s interesting is how the water behaves when it is disturbed. Its movement changes the centrifugal force on it, as Coriolis’s equations predicted, so that when heated from below, it moves not in huge convection currents but up and down in narrow rotating columns. This discovery led the geophysicists Walter Elsasser and Edward Bullard to realise that the same forces would deflect convection currents in Earth’s metal outer core that are driven by radioactive decay. They are diverted into north-to-south columns of rotating metal that act like self-excited dynamos, producing the magnetic field that shields Earth from charged particles. A simple laboratory demonstration had illuminated events in Earth’s core that had been hidden in Coriolis’s equations.

Today, perhaps the most damaging failure to translate the mathematics of spin into easy-to-grasp concepts is in the fields of biomechanics and sports science. Our bodies are complex systems of rotating joints, but despite the sophistication of modern motion analysis software, few researchers realise that accelerating our joints can produce torques that actively accelerate our limbs. Biomechanics researchers are only starting to realise that accelerating our bodies upwards at the start of each step swings our arms and legs when we walk, and that a sling action straightens them at the end of each step.

In the same way, when we throw things, we use a multi-stage sling action; rotating our shoulders accelerates first our upper arm, then our forearm and finally our hands. And the reason we can wield heavy sledgehammers and swing wooden clubs to smash golf balls down the fairway is that their handles act as further sling elements; they accelerate forwards due to the centrifugal forces on them without us having to flex our wrists. Failing to articulate these simple mechanical concepts has made biomechanics ill-equipped to communicate with and help physiotherapists, sports coaches and roboticists.

And there is still confusion about the simplest aspects of rotation among physicists. Even Richard Feynman, for instance, was unable to explain the so-called Dzhanibekov effect – why spinning wing nuts on the International Space Station flip every few seconds. This was despite the fact that the mathematician Leonhard Euler had shown this should happen almost 300 years ago. The same is also true of more down-to-earth events: how children power playground swings and how cats land on their feet, for example.

The truth is that the basics of physics, despite involving simple mathematics, are harder to grasp than we tend to think. It took me two years, for instance, to master just the science of spin and rotation for my latest book. We need to spend more time thinking about, visualising and demonstrating basic physical concepts. If we do, we could produce a generation of physicists who can communicate better with everyone else and discover more about the world around us. The answers are probably already there, hidden in the equations.

The Science of Spin by Roland Ennos is out now.

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*Credit for article given to Roland Ennos*

Mathematicians are stunned by the discovery that prime numbers are pickier than previously thought. The find suggests number theorists need to be a little more careful when exploring the vast infinity of primes.

Primes, the numbers divisible only by themselves and 1, are the building blocks from which the rest of the number line is constructed, as all other numbers are created by multiplying primes together. That makes deciphering their mysteries key to understanding the fundamentals of arithmetic.

Although whether a number is prime or not is pre-determined, mathematicians don’t have a way to predict which numbers are prime, and so tend to treat them as if they occur randomly. Now Kannan Soundararajan and Robert Lemke Oliver of Stanford University in California have discovered that isn’t quite right.

“It was very weird,” says Soundararajan. “It’s like some painting you are very familiar with, and then suddenly you realise there is a figure in the painting you’ve never seen before.”

Surprising order

So just what has got mathematicians spooked? Apart from 2 and 5, all prime numbers end in 1, 3, 7 or 9 – they have to, else they would be divisible by 2 or 5 – and each of the four endings is equally likely. But while searching through the primes, the pair noticed that primes ending in 1 were less likely to be followed by another prime ending in 1. That shouldn’t happen if the primes were truly random – consecutive primes shouldn’t care about their neighbour’s digits.

“In ignorance, we thought things would be roughly equal,” says Andrew Granville of the University of Montreal, Canada. “One certainly believed that in a question like this we had a very strong understanding of what was going on.”

The pair found that in the first hundred million primes, a prime ending in 1 is followed by another ending in 1 just 18.5 per cent of the time. If the primes were distributed randomly, you’d expect to see two 1s next to each other 25 per cent of the time. Primes ending in 3 and 7 take up the slack, each following a 1 in 30 per cent of primes, while a 9 follows a 1 in around 22 per cent of occurrences.

Similar patterns showed up for the other combinations of endings, all deviating from the expected random values. The pair also found them in other bases, where numbers are counted in units other than 10s. That means the patterns aren’t a result of our base-10 numbering system, but something inherent to the primes themselves. The patterns become more in line with randomness as you count higher – the pair have checked up to a few trillion – but still persists.

“I was very surprised,” says James Maynard of the University of Oxford, UK, who on hearing of the work immediately performed his own calculations to check the pattern was there. “I somehow needed to see it for myself to really believe it.”

Stretching to infinity

Thankfully, Soundararajan and Lemke Oliver think they have an explanation. Much of the modern research into primes is underpinned G H Hardy and John Littlewood, two mathematicians who worked together at the University of Cambridge in the early 20^th century. They came up with a way to estimate how often pairs, triples and larger grouping of primes will appear, known as the k-tuple conjecture.

Just as Einstein’s theory of relativity is an advance on Newton’s theory of gravity, the Hardy-Littlewood conjecture is essentially a more complicated version of the assumption that primes are random – and this latest find demonstrates how the two assumptions differ. “Mathematicians go around assuming primes are random, and 99 per cent of the time this is correct, but you need to remember the 1 per cent of the time it isn’t,” says Maynard.

The pair used Hardy and Littlewood’s work to show that the groupings given by the conjecture are responsible for introducing this last-digit pattern, as they place restrictions on where the last digit of each prime can fall. What’s more, as the primes stretch to infinity, they do eventually shake off the pattern and give the random distribution mathematicians are used to expecting.

“Our initial thought was if there was an explanation to be found, we have to find it using the k-tuple conjecture,” says Soundararajan. “We felt that we would be able to understand it, but it was a real puzzle to figure out.”

The k-tuple conjecture is yet to be proven, but mathematicians strongly suspect it is correct because it is so useful in predicting the behaviour of the primes. “It is the most accurate conjecture we have, it passes every single test with flying colours,” says Maynard. “If anything I view this result as even more confirmation of the k-tuple conjecture.”

Although the new result won’t have any immediate applications to long-standing problems about primes like the twin-prime conjecture or the Riemann hypothesis, it has given the field a bit of a shake-up. “It gives us more of an understanding, every little bit helps,” says Granville. “If what you take for granted is wrong, that makes you rethink some other things you know.”

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*Credit for article given to Jacob Aron*